Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

Alessio Figalli; Vito Mandorino

doi:10.3934/dcds.2011.31.1325

Article Contents

2011, Volume 31, Issue 4: 1325-1346. Doi: 10.3934/dcds.2011.31.1325 open access

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Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

Alessio Figalli^1, and
Vito Mandorino^2,

1.
University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202
2.
Université Paris-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16

Received: October 31, 2009

Revised: July 31, 2010

Published: September 2011

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Abstract

In this paper we study the properties of curves minimizing mechanical Lagrangianwhere the potential is Sobolev. Since a Sobolev functionis only defined almost everywhere, no pointwise results can be obtained in this framework,and our point of view is shifted from single curves to measures in the space of paths.This study is motived by the goal of understanding the properties ofvariational solutions to the incompressible Euler equations.

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Mathematics Subject Classification: Primary: 49Kxx, 49Lxx; Secondary: 70H03, 37J50.

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